arXiv subjects

Sam Hadden

Publications and source records attributed to Sam Hadden.

At least 19 recordsLinked to original sources

arXiv2026

The Persistent Missing Mass Problem in Planet Formation

Recent ground-based microlensing surveys suggest that our Galaxy may abound with small free floating planets, potentially up to $\sim$21 such planets per star. We explore the implication of such possibility on the mass budget for planet formation. When the microlensing planets, both bound and free-floating, are taken into account, along with the short-period planets, T Tauri disks have insufficient mass to source the mass of known planets, even if all the solids convert into planetary bodies. Younger Class 0/I disks can help resolve the problem but generally fall short of the required mass when variable planet formation efficiency from pebble or planetesimal accretion is taken into consideration. If the free-floating planet mass function is as bottom-heavy as reported, heavier Class 0/I disks may be necessary. Alternatively, free-floaters may preferentially form in the most massive disks around massive stars consuming the majority of the mass budget, leading to a decrease in the bound planet occurrence rate for higher mass stars, which is observed. Precise constraints on the bottom of planet mass function are necessary: a peaked mass function may eliminate the missing mass problem; by contrast, verifying a bottom-heavy function could spell a crisis in planet formation.

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arXiv2026

Rattle-and-Break: the Impact of Planetesimal Scattering on Super-Earth Resonant Chains

The spacings of super-Earths in multi-transiting systems exhibit a distribution that is broad and mostly featureless, with the exception of notable excesses of planet pairs situated a few percent wide of first-order mean motion resonances (MMRs). In this work, we extend the so-called "breaking-the-chains" model to account for both of these characteristics. Assuming that super-Earths are settled into stable chains of resonances after disk-driven migration, we show that scattering a planetesimal population that contains only a few percent of a system's mass can reorganize primordial chains in remarkable ways. The planetesimal scattering "rattles" the chains by repelling adjacent planet pairs wide of their initial MMRs. Some chains remain rattled but otherwise intact and make up the observed excesses wide of MMRs. In other systems, however, this initial rattling sows the seeds of later orbital instabilities that break the chains entirely. If individual planetesimals' masses are of order a Pluto mass or so, the onset of these instabilities can occur tens or hundreds of Myr after birth, naturally explaining the apparent disappearance of near-resonant pairs on this timescale. The origin of such Pluto-mass debris is currently unknown.

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arXiv2025

Free Floating or Merely Detached?

Microlensing surveys suggest the presence of a surprisingly large population of free-floating planets, with a rate of about two Neptunes per star. The origin of such objects is not known, neither do we know if they are truly unbound or are merely orbiting at large separations from their host stars. Here, we investigate planet-planet scattering as a possible origin through numerical simulations of unstable multi-planet systems. We find that planet ejection by scattering can be slow, often taking more than billions of years for Neptune-mass scatterers orbiting at a few AU and beyond. Moreover, this process invariably delivers planets to orbits of hundreds of AU that are protected from further scattering. We call these ``detached" planets. Under the scattering hypothesis, we estimate that about half of the reported ``free-floating" Neptunes are not free but merely ``detached".

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arXiv2024

Eccentricities of Close Stellar Binaries

Orbits of stellar binaries are in general eccentric. These eccentricities encode information about their early lives. Here, we use thousands of main-sequence binaries from the Gaia DR3 catalog to reveal that, binaries inwards of a few AU exhibit a simple Rayleigh distribution with a mode sigma_e ~ 0.3. We find the same distribution for binaries from M to A spectral types, and from tens of days to a thousand days (possibly extending to tens of AU). This observed distribution is most likely primordial and its invariance suggests a single universal process. One possibility is eccentricity excitation by circumbinary disks. Another, as is suggested by the Rayleigh form, is weak scattering and ejection of brown-dwarf objects. We explore this latter scenario and find that the binary eccentricities reach an equi-partition value of sigma_e ~ sqrt{M_bd M_*}. So to explain the observed mode, the brown dwarfs will have to be of order one tenth the stellar masses, and be at least as abundant in the Galaxy as the close binaries. The veracity of both proposals remains to be tested.

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arXiv2024

Influence of Modeling Assumptions on the Inferred Dynamical State of Resonant Systems: A Case Study of the HD 45364 System

Planetary systems exhibiting mean-motion resonances (MMRs) offer unique opportunities to study the imprint of disk-induced migration on the orbital architectures of planetary systems. The HD 45364 system, discovered via the radial velocity (RV) method to host two giant planets in a 3:2 MMR, has been the subject of several studies attempting to reconstruct the system's orbital migration history based on its present-day resonant configuration. Recently, Li et al. (2022) called into question the system's residence in the 3:2 MMR based on a revised orbital solution derived from an expanded set of RV observations that extend the time baseline of the original discovery data by over a decade. However, we show that inferences about the planets' dynamical state with respect to the 3:2 MMR are sensitive to the particular prior assumptions adopted in the orbital modeling. Using $N$-body dynamical models, we show that orbital solutions constrained to reside deep in the 3:2 MMR fit the RV data with a similar quality to unconstrained orbital solutions. We conclude that the RV observations of HD 45364 are consistent with orbital configurations produced by smooth migration and resonance capture. We further show that past convergent orbital migration can reproduce the system's present-day orbital configuration provided that the ratio of migration to eccentricity damping timescales, $K$, was in the range $11\lesssim K \lesssim 144$. We also find that dynamical interactions in the system can break the usual mass-inclination degeneracy inherent to Keplerian models of RV observations and constrain the planets' absolute masses to within a factor of $\sim1.5$.

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arXiv2024

Accelerating Giant Impact Simulations with Machine Learning

Constraining planet formation models based on the observed exoplanet population requires generating large samples of synthetic planetary systems, which can be computationally prohibitive. A significant bottleneck is simulating the giant impact phase, during which planetary embryos evolve gravitationally and combine to form planets, which may themselves experience later collisions. To accelerate giant impact simulations, we present a machine learning (ML) approach to predicting collisional outcomes in multiplanet systems. Trained on more than 500,000 $N$-body simulations of three-planet systems, we develop an ML model that can accurately predict which two planets will experience a collision, along with the state of the post-collision planets, from a short integration of the system's initial conditions. Our model greatly improves on non-ML baselines that rely on metrics from dynamics theory, which struggle to accurately predict which pair of planets will experience a collision. By combining with a model for predicting long-term stability, we create an ML-based giant impact emulator, which can predict the outcomes of giant impact simulations with reasonable accuracy and a speedup of up to four orders of magnitude. We expect our model to enable analyses that would not otherwise be computationally feasible. As such, we release our training code, along with an easy-to-use API for our collision outcome model and giant impact emulator.

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arXiv2024

Action-Angle Variables for Axisymmetric Potentials via Birkhoff Normalization

We describe a method for calculating action-angle variables in axisymmetric galactic potentials using Birkhoff normalization, a technique from Hamiltonian perturbation theory. An advantageous feature of this method is that it yields explicit series expressions for both the forward and inverse transformations between the action-angle variables and position-velocity data. It also provides explicit expressions for the Hamiltonian and dynamical frequencies as functions of the action variables. We test this method by examining orbits in a Miyamoto-Nagai model potential and compare it to the popular St\"ackel approximation method. When vertical actions are not too large, the Birkhoff normalization method achieves fractional errors smaller than a part in $10^{3}$ and outperforms the St\"ackel approximation. We also show that the range over which Birkhoff normalization provides accurate results can be extended by constructing Pad\'e approximants from the perturbative series expressions developed with the method. Numerical routines in Python for carrying out the Birkhoff normalization procedure are made available.

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arXiv2024

The instability mechanism of compact multiplanet systems

To improve our understanding of orbital instabilities in compact planetary systems, we compare suites of $N$-body simulations against numerical integrations of simplified dynamical models. We show that, surprisingly, dynamical models that account for small sets of resonant interactions between the planets can accurately recover $N$-body instability times. This points toward a simple physical picture in which a handful of three-body resonances, generated by interactions between nearby two-body mean motion resonances, overlap and drive chaotic diffusion, leading to instability. Motivated by this, we show that instability times are well described by a power law relating instability time to planet separations, measured in units of fractional semi-major axis difference divided by the planet-to-star mass ratio to the $1/4$ power, rather than the frequently adopted $1/3$ power implied by measuring separations in units of mutual Hill radii. For idealized systems, the parameters of this power-law relationship depend only on the ratio of the planets' orbital eccentricities to the orbit-crossing value, and we report an empirical fit to enable quick instability time predictions. This relationship predicts that observed systems comprised of three or more sub-Neptune-mass planets must be spaced with period ratios $P \gtrsim 1.35$ and that tightly spaced systems ($P \lesssim 1.5$) must possess very low eccentricities ($e \lesssim 0.05$) to be stable for more than $10^9$ orbits.

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arXiv2023

Masses, Revised Radii, and a Third Planet Candidate in the "Inverted" Planetary System Around TOI-1266

Is the population of close-in planets orbiting M dwarfs sculpted by thermally driven escape or is it a direct outcome of the planet formation process? A number of recent empirical results strongly suggest the latter. However, the unique architecture of the TOI-1266 system presents a challenge to models of planet formation and atmospheric escape given its seemingly "inverted" architecture of a large sub-Neptune ($P_b=10.9$ days, $R_{p,b}=2.62\pm 0.11\, \mathrm{R}_{\oplus}$) orbiting interior to that of the system's smaller planet ($P_c=18.8$ days, $R_{p,c}=2.13\pm 0.12\, \mathrm{R}_{\oplus}$). Here we present revised planetary radii based on new TESS and diffuser-assisted ground-based transit observations, and characterize both planetary masses using a set of 145 radial velocity measurements from HARPS-N ($M_{p,b}=4.23\pm 0.69\, \mathrm{M}_{\oplus}, M_{p,c}=2.88\pm 0.80\, \mathrm{M}_{\oplus}$). Our analysis also reveals a third planet candidate ($P_d=32.3$ days, $M_{p,d}\sin{i} = 4.59^{+0.96}_{-0.94}\, \mathrm{M}_{\oplus}$), which if real, would form a chain of near 5:3 period ratios, although the system is likely not in a mean motion resonance. Our results indicate that TOI-1266 b and c are among the lowest density sub-Neptunes around M dwarfs and likely exhibit distinct bulk compositions of a gas-enveloped terrestrial ($X_{\mathrm{env},b}=5.5\pm 0.7$%) and a water-rich world (WMF$_c=59\pm 14$%), which is supported by hydrodynamic escape models. If distinct bulk compositions are confirmed through atmospheric characterization, the system's unique architecture would represent an interesting test case of inside-out sub-Neptune formation at pebble traps.

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arXiv2023

Scattered Disk Dynamics: The Mapping Approach

We derive, and discuss the properties of, a symplectic map for the dynamics of bodies on nearly parabolic orbits. The orbits are perturbed by a planet on a circular, coplanar orbit interior to the pericenter of the parabolic orbit. The map shows excellent agreement with direct numerical integrations and elucidates how the dynamics depends on perturber mass and pericenter distance. We also use the map to explore the onset of chaos, statistical descriptions of chaotic transport, and sticking in mean-motion resonances. We discuss implications of our mapping model for the dynamical evolution of the solar system's scattered disk and other highly eccentric trans-Neptunian objects.

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arXiv2023

Mercury's chaotic secular evolution as a subdiffusive process

Mercury's orbit can destabilize, generally resulting in a collision with either Venus or the Sun. Chaotic evolution can cause g1 to decrease to the approximately constant value of g5 and create a resonance. Previous work has approximated the variation in g1 as stochastic diffusion, which leads to a phenomological model that can reproduce the Mercury instability statistics of secular and N-body models on timescales longer than 10 Gyr. Here we show that the diffusive model underpredicts the Mercury instability probability by a factor of 3-10,000 on timescales less than 5 Gyr, the remaining lifespan of the Solar System. This is because g1 exhibits larger variations on short timescales than the diffusive model would suggest. To better model the variations on short timescales, we build a new subdiffusive phenomological model for g1. Subdiffusion is similar to diffusion but exhibits larger displacements on short timescales and smaller displacements on long timescales. We choose model parameters based on the behavior of the g1 trajectories in the N-body simulations, leading to a tuned model that can reproduce Mercury instability statistics from 1-40 Gyr. This work motivates fundamental questions in Solar System dynamics: Why does subdiffusion better approximate the variation in g1 than standard diffusion? Why is there an upper bound on g1, but not a lower bound that would prevent it from reaching g5?

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arXiv2023

Intra-system uniformity: a natural outcome of dynamical sculpting

There is evidence that exoplanet systems display intra-system uniformity in mass, radius, and orbital spacing (like ``peas in a pod'') when compared with the system-to-system variations of planetary systems. This has been interpreted as the outcome of the early stages of planet formation, indicative of a picture in which planets form at characteristic mass scales with uniform separations. In this paper, we argue instead that intra-system uniformity in planet sizes and orbital spacings likely arose from the dynamical sculpting of initially-overly-packed planetary systems (in other words, the giant impact phase). With a suite of $N$-body simulations, we demonstrate that systems with random initial masses and compact planet spacings naturally develop intra-system uniformity, in quantitative agreement with observations, due to collisions between planets. Our results suggest that the pre-giant impact planet mass distribution is fairly wide and provide evidence for the prevalence of dynamical sculpting in shaping the observed population of exoplanets.

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arXiv2023

Self-Consistent Spin, Tidal and Dynamical Equations of Motion in the REBOUNDx Framework

We have introduced self-consistent spin, tidal and dynamical equations of motion into REBOUNDx, a library of additional effects for the popular N-body integrator REBOUND. The equations of motion used are derived from the constant time lag approximation to the equilibrium tide model of tidal friction. These effects will allow the study of a variety of systems where the full dynamical picture cannot be encapsulated by point particle dynamics. We provide several test cases and benchmark the code's performance against analytic predictions. The open-source code is available in the most recent release of REBOUNDx.

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arXiv2022

Simple physics and integrators accurately reproduce Mercury instability statistics

The long-term stability of the Solar System is an issue of significant scientific and philosophical interest. The mechanism leading to instability is Mercury's eccentricity being pumped up so high that Mercury either collides with Venus or is scattered into the Sun. Previously, only three five-billion-year $N$-body ensembles of the Solar System with thousands of simulations have been run to assess long-term stability. We generate two additional ensembles, each with 2750 members, and make them publicly available at \texttt{https://archive.org/details/@dorianabbot}. We find that accurate Mercury instability statistics can be obtained by (1) including only the Sun and the 8 planets, (2) using a simple Wisdom-Holman scheme without correctors, (3) using a basic representation of general relativity, and (4) using a time step of 3.16 days. By combining our Solar System ensembles with previous ensembles we form a 9,601-member ensemble of ensembles. In this ensemble of ensembles, the logarithm of the frequency of a Mercury instability event increases linearly with time between 1.3 and 5 Gyr, suggesting that a single mechanism is responsible for Mercury instabilities in this time range and that this mechanism becomes more active as time progresses. Our work provides a robust estimate of Mercury instability statistics over the next five billion years, outlines methodologies that may be useful for exoplanet system investigations, and provides two large ensembles of publicly available Solar System integrations that can serve as testbeds for theoretical ideas as well as training sets for artificial intelligence schemes.

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arXiv2022

celmech: A Python package for celestial mechanics

We present celmech, an open-source Python package designed to facilitate a wide variety of celestial mechanics calculations. The package allows users to formulate and integrate equations of motion incorporating user-specified terms from the classical disturbing function expansion of the interaction potential between pairs of planets. The code can be applied, for example, to isolate the contribution of particular resonances to a system's dynamical evolution and develop simple analytical models with the minimum number of terms required to capture a particular dynamical phenomenon. Equations and expressions can be easily manipulated by leveraging the extensive symbolic mathematics capabilities of the sympy Python package. The celmech package is designed to interface seamlessly with the popular $N$-body code REBOUND to facilitate comparisons between calculation results and direct $N$-body integrations. The code is extensively documented and numerous example Jupyter notebooks illustrating its use are available online.

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arXiv2022

The Effects of Disk Induced Apsidal Precession on Planets Captured into Mean Motion Resonance

The process of migration into resonance capture has been well studied for planetary systems where the gravitational potential is generated exclusively by the star and planets. However, massive protoplanetary disks add a significant perturbation to these models. In this paper we consider two limiting cases of disk-induced precession on migrating planets and find that small amounts of precession significantly affect the equilibrium reached by migrating planets. We investigate these effects with a combination of semi-analytic models of the resonance and numerical integrations. We also consider the case of the disk's dispersal, which can excite significant libration amplitude and can cause ejection from resonance for large enough precession rates. Both of these effects have implications for interpreting the known exoplanet population and may prove to be important considerations as the population of well-characterized exoplanet systems continues to grow.

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arXiv2021

Stepsize errors in the $N$-body problem: discerning Mercury's true possible long-term orbits

Numerical integrations of the Solar System have been carried out for decades. Their results have been used, for example, to determine whether the Solar System is chaotic, whether Mercury's orbit is stable, or to help discern Earth's climate history. We argue that all of the past studies we consider in this work are affected by numerical chaos to different degrees, affecting the possible orbits and instability probability of Mercury, sometimes significantly. We show how to eliminate the effects of numerical chaos by resolving Mercury's pericentre passage. We also show that several higher order symplectic maps do not exhibit significant differences in resolving pericentre passage of Mercury (at fixed time step), making their advantages suspect for calculating long-term orbits. Resolving pericentre passage affects a wide array of orbital numerical studies, like exoplanet studies, studies of the galactic centre, and other $N$-body problems.

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arXiv2021

The Criterion for Chaos in Three-Planet Systems

We establish the criterion for chaos in three-planet systems, for systems similar to those discovered by the Kepler spacecraft. Our main results are as follows: (i) The simplest criterion, which is based on overlapping mean motion resonances MMR's), only agrees with numerical simulations at a very crude level. (ii) Much greater accuracy is attained by considering neighboring MMR's that do not overlap. We work out the width of the chaotic zones around each of the neighbors, and also provide simple approximate expressions for the widths. (iii) Even greater accuracy is provided by the overlap of three-body resonances (3BR's), which accounts for fine-grained structure seen in maps from N-body simulations, and also predicts the Lyapunov times. Previous studies conflict on whether overlap of MMR's or of 3BR's drive interplanetary chaos. We show that both do, and in fact they are merely different ways of looking at the same effect. (iv) We compare both criteria with high-resolution maps of chaos from N-body simulations, and show that they agree at a high level of detail.

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